46:, namely, there exist equilibrium states which can be represented by points on two-dimensional surface (manifold) and the distance between these equilibrium states is related to the fluctuation between them. This concept is associated to probabilities, i.e. the less probable a fluctuation between states, the further apart they are. This can be recognized if one considers the
466:
It has long been observed that the
Ruppeiner metric is flat for systems with noninteracting underlying statistical mechanics such as the ideal gas. Curvature singularities signal critical behaviors. In addition, it has been applied to a number of statistical systems including
377:+ ...) in differential form with a few manipulations. The Weinhold geometry is also considered as a thermodynamic geometry. It is defined as a Hessian of the internal energy with respect to entropy and other extensive parameters.
665:
of the black hole. Calculating the
Ruppeiner geometry of the black hole's entropy is, in principle, straightforward, but it is important that the entropy should be written in terms of extensive parameters,
249:
461:
557:
356:
140:
281:. For small systems (systems where fluctuations are large), the Ruppeiner metric may not exist, as second derivatives of the entropy are not guaranteed to be non-negative.
715:
587:
812:
738:
659:
635:
611:
944:
168:
383:
638:
487:
in higher dimensions, where the curvature singularity signals thermodynamic instability, as found earlier by conventional methods.
761:
365:
is the temperature of the system under consideration. Proof of the conformal relation can be easily done when one writes down the
499:
491:
294:
949:
934:
366:
64:
908:Ă…man, John E.; Bengtsson, Ingemar; Pidokrajt, Narit; Ward, John (2008). "Thermodynamic Geometries of Black Holes".
266:
refers to the extensive parameters of the system. Mathematically, the
Ruppeiner geometry is one particular type of
43:
480:
278:
939:
277:
The
Ruppeiner metric can be understood as the thermodynamic limit (large systems limit) of the more general
468:
35:
can be represented by
Riemannian geometry, and that statistical properties can be derived from the model.
271:
887:
853:
776:
signature. This calculation cannot be done for the
Schwarzschild black hole, because its entropy is
267:
32:
20:
24:
672:
565:
843:
590:
913:
895:
861:
782:
155:
773:
484:
285:
259:
483:, with some physically relevant results. The most physically significant case is for the
891:
857:
38:
This geometrical model is based on the inclusion of the theory of fluctuations into the
769:
723:
644:
620:
614:
596:
28:
928:
878:
Ruppeiner, George (1995). "Riemannian geometry in thermodynamic fluctuation theory".
765:
757:
662:
47:
764:, the Ruppeiner metric has a Lorentzian signature which corresponds to the negative
865:
917:
899:
58:
in the distance formula (line element) between the two equilibrium states
741:
159:
39:
848:
471:
gas. Recently the anyon gas has been studied using this approach.
244:{\displaystyle g_{ij}^{R}=-\partial _{i}\partial _{j}S(U,N^{a})}
456:{\displaystyle g_{ij}^{W}=\partial _{i}\partial _{j}U(S,N^{a})}
756:. The signature of the metric reflects the sign of the hole's
834:
552:{\displaystyle S={\frac {k_{\text{B}}c^{3}A}{4G\hbar }}}
31:. George Ruppeiner proposed it in 1979. He claimed that
490:
The entropy of a black hole is given by the well-known
785:
726:
675:
647:
623:
599:
568:
502:
386:
351:{\displaystyle ds_{R}^{2}={\frac {1}{T}}ds_{W}^{2}\,}
297:
171:
67:
284:The Ruppeiner metric is conformally related to the
806:
732:
709:
653:
629:
605:
581:
551:
455:
350:
243:
134:
154:is the symmetric metric tensor which is called a
135:{\displaystyle ds^{2}=g_{ij}^{R}dx^{i}dx^{j},\,}
8:
847:
784:
725:
698:
674:
646:
622:
598:
573:
567:
526:
516:
509:
501:
444:
422:
412:
399:
391:
385:
347:
341:
336:
319:
310:
305:
296:
232:
210:
200:
184:
176:
170:
131:
122:
109:
96:
88:
75:
66:
826:
543:
158:, defined as a negative Hessian of the
910:The Eleventh Marcel Grossmann Meeting
817:which renders the metric degenerate.
19:is thermodynamic geometry (a type of
7:
479:This geometry has been applied to
419:
409:
207:
197:
14:
639:Newtonian constant of gravitation
475:Application to black hole systems
274:used in mathematical statistics.
145:where the matrix of coefficients
945:New College of Florida faculty
801:
795:
748:are the conserved charges and
704:
685:
450:
431:
238:
219:
1:
866:10.1103/PhysRevLett.99.100602
762:Reissner–Nordström black hole
710:{\displaystyle S=S(M,N^{a})}
582:{\displaystyle k_{\text{B}}}
367:first law of thermodynamics
966:
918:10.1142/9789812834300_0182
768:it possess, while for the
492:Bekenstein–Hawking formula
44:equilibrium thermodynamics
900:10.1103/RevModPhys.67.605
880:Reviews of Modern Physics
481:black hole thermodynamics
279:Fisher information metric
270:and it is similar to the
262:(mass) of the system and
23:) using the language of
912:. pp. 1511–1513.
808:
807:{\displaystyle S=S(M)}
744:of the black hole and
734:
711:
655:
631:
607:
583:
553:
457:
352:
245:
136:
809:
735:
712:
656:
632:
608:
584:
554:
458:
353:
246:
137:
33:thermodynamic systems
950:Mathematical physics
783:
724:
673:
645:
621:
597:
566:
500:
384:
295:
268:information geometry
169:
65:
21:information geometry
935:Riemannian geometry
892:1995RvMP...67..605R
858:2007PhRvL..99j0602C
661:is the area of the
404:
346:
315:
189:
101:
25:Riemannian geometry
804:
730:
707:
651:
627:
603:
591:Boltzmann constant
579:
549:
453:
387:
348:
332:
301:
241:
172:
132:
84:
17:Ruppeiner geometry
733:{\displaystyle M}
654:{\displaystyle A}
630:{\displaystyle G}
606:{\displaystyle c}
576:
547:
519:
327:
272:Fisher–Rao metric
957:
921:
903:
870:
869:
851:
831:
813:
811:
810:
805:
739:
737:
736:
731:
716:
714:
713:
708:
703:
702:
660:
658:
657:
652:
636:
634:
633:
628:
612:
610:
609:
604:
588:
586:
585:
580:
578:
577:
574:
558:
556:
555:
550:
548:
546:
535:
531:
530:
521:
520:
517:
510:
462:
460:
459:
454:
449:
448:
427:
426:
417:
416:
403:
398:
357:
355:
354:
349:
345:
340:
328:
320:
314:
309:
250:
248:
247:
242:
237:
236:
215:
214:
205:
204:
188:
183:
156:Ruppeiner metric
141:
139:
138:
133:
127:
126:
114:
113:
100:
95:
80:
79:
965:
964:
960:
959:
958:
956:
955:
954:
925:
924:
907:
877:
874:
873:
836:Phys. Rev. Lett
833:
832:
828:
823:
781:
780:
752:runs from 1 to
722:
721:
694:
671:
670:
643:
642:
619:
618:
595:
594:
569:
564:
563:
536:
522:
512:
511:
498:
497:
485:Kerr black hole
477:
440:
418:
408:
382:
381:
293:
292:
286:Weinhold metric
260:internal energy
228:
206:
196:
167:
166:
153:
118:
105:
71:
63:
62:
57:
12:
11:
5:
963:
961:
953:
952:
947:
942:
940:Thermodynamics
937:
927:
926:
923:
922:
905:
886:(3): 605–659.
872:
871:
825:
824:
822:
819:
815:
814:
803:
800:
797:
794:
791:
788:
770:BTZ black hole
729:
718:
717:
706:
701:
697:
693:
690:
687:
684:
681:
678:
650:
626:
615:speed of light
602:
572:
560:
559:
545:
542:
539:
534:
529:
525:
515:
508:
505:
476:
473:
464:
463:
452:
447:
443:
439:
436:
433:
430:
425:
421:
415:
411:
407:
402:
397:
394:
390:
359:
358:
344:
339:
335:
331:
326:
323:
318:
313:
308:
304:
300:
252:
251:
240:
235:
231:
227:
224:
221:
218:
213:
209:
203:
199:
195:
192:
187:
182:
179:
175:
149:
143:
142:
130:
125:
121:
117:
112:
108:
104:
99:
94:
91:
87:
83:
78:
74:
70:
53:
29:thermodynamics
13:
10:
9:
6:
4:
3:
2:
962:
951:
948:
946:
943:
941:
938:
936:
933:
932:
930:
919:
915:
911:
906:
901:
897:
893:
889:
885:
881:
876:
875:
867:
863:
859:
855:
850:
845:
841:
837:
830:
827:
820:
818:
798:
792:
789:
786:
779:
778:
777:
775:
771:
767:
766:heat capacity
763:
759:
758:specific heat
755:
751:
747:
743:
727:
699:
695:
691:
688:
682:
679:
676:
669:
668:
667:
664:
663:event horizon
648:
640:
624:
616:
600:
592:
570:
540:
537:
532:
527:
523:
513:
506:
503:
496:
495:
494:
493:
488:
486:
482:
474:
472:
470:
469:Van der Waals
445:
441:
437:
434:
428:
423:
413:
405:
400:
395:
392:
388:
380:
379:
378:
376:
372:
368:
364:
342:
337:
333:
329:
324:
321:
316:
311:
306:
302:
298:
291:
290:
289:
287:
282:
280:
275:
273:
269:
265:
261:
257:
233:
229:
225:
222:
216:
211:
201:
193:
190:
185:
180:
177:
173:
165:
164:
163:
161:
157:
152:
148:
128:
123:
119:
115:
110:
106:
102:
97:
92:
89:
85:
81:
76:
72:
68:
61:
60:
59:
56:
52:
49:
48:metric tensor
45:
41:
36:
34:
30:
26:
22:
18:
909:
883:
879:
839:
835:
829:
816:
772:, we have a
753:
749:
745:
719:
561:
489:
478:
465:
374:
370:
362:
360:
283:
276:
263:
255:
253:
150:
146:
144:
54:
50:
37:
16:
15:
929:Categories
842:: 100602.
821:References
849:0706.0559
774:Euclidean
544:ℏ
420:∂
410:∂
208:∂
198:∂
194:−
162:function
27:to study
760:. For a
742:ADM mass
888:Bibcode
854:Bibcode
637:is the
613:is the
589:is the
258:is the
160:entropy
720:where
562:where
361:where
254:where
40:axioms
844:arXiv
641:and
288:via
914:doi
896:doi
862:doi
740:is
375:TdS
42:of
931::
894:.
884:67
882:.
860:.
852:.
840:99
838:.
617:,
593:,
373:=
371:dU
151:ij
55:ij
920:.
916::
904:.
902:.
898::
890::
868:.
864::
856::
846::
802:)
799:M
796:(
793:S
790:=
787:S
754:n
750:a
746:N
728:M
705:)
700:a
696:N
692:,
689:M
686:(
683:S
680:=
677:S
649:A
625:G
601:c
575:B
571:k
541:G
538:4
533:A
528:3
524:c
518:B
514:k
507:=
504:S
451:)
446:a
442:N
438:,
435:S
432:(
429:U
424:j
414:i
406:=
401:W
396:j
393:i
389:g
369:(
363:T
343:2
338:W
334:s
330:d
325:T
322:1
317:=
312:2
307:R
303:s
299:d
264:N
256:U
239:)
234:a
230:N
226:,
223:U
220:(
217:S
212:j
202:i
191:=
186:R
181:j
178:i
174:g
147:g
129:,
124:j
120:x
116:d
111:i
107:x
103:d
98:R
93:j
90:i
86:g
82:=
77:2
73:s
69:d
51:g
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.